Babson–Kozlov vanishing conjecture for odd-cycle Hom-complexes

Let $C_{2r+1}$ be an odd cycle, let $K_n$ be the complete graph on $n$ vertices, and let $\varpi_1^{n-2}(\text{\tt Hom}(C_{2r+1},K_n))$ denote the indicated Stiefel-Whitney characteristic class. Babson–Kozlov vanishing conjecture. For every positive integer $r$ and every $n\geq 2$, one has

$$\varpi_1^{n-2}(\text{\tt Hom}(C_{2r+1},K_n))=0.$$

The source records proofs for $r=1$ and for odd $n$, as well as the case $(r,n)=(2,4)$, but leaves the general claim unresolved.